M/G/1 Queue with State Dependent Service Times

Abstract

In this paper we study the state dependent M/G/1 queueing system in which the service time can change at departure epochs. The model is a special case of an already investigated model. As a result of the narrowed scope we get numerically more effective and closed form solutions. We provide the steady-state distribution of the number of customers in the system and the stability condition, both in terms of quantities computed by recursions. We also study the model with finite number of state dependent service time distributions. For this model variant, closed form expressions are provided for the probability-generating function and the mean of the steady-state number of customers, which are computed from a system of linear equations. Finally we also investigate the model with state dependent linear interpolation of two service times. For this model, we derive an explicit expression for the probability generating function of the steady-state number of customers and establish a simple, explicit stability condition. This model behaviour implements a control of number of customers in the system.